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Containing and shrinking ellipsoids in the path-following algorithm

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Abstract

We describe a new potential function and a sequence of ellipsoids in the path-following algorithm for convex quadratic programming. Each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors. Furthermore, the volumes of the ellipsoids shrink at the ratio\(2^{ - \Omega (\sqrt n )} \), in comparison to 2−Ω(1) in Karmarkar's algorithm and 2−Ω(1/n) in the ellipsoid method. We also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming.

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Authors and Affiliations

  1. Department of Management Sciences, The Univeristy of Iowa, Iowa City, IA, USA

    Yinyu Ye

  2. Department of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, USA

    Michael J. Todd

Authors
  1. Yinyu Ye
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  2. Michael J. Todd
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Additional information

Research supported by The U.S. Army Research Office through The Mathematical Sciences Institute of Cornell University when the author was visiting at Cornell.

Research supported in part by National Science Foundation Grant ECS-8602534 and Office of Naval Research Contract N00014-87-K-0212.

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Ye, Y., Todd, M.J. Containing and shrinking ellipsoids in the path-following algorithm. Mathematical Programming 47, 1–9 (1990). https://doi.org/10.1007/BF01580848

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  • DOI: https://doi.org/10.1007/BF01580848

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